Newspace parameters
| Level: | \( N \) | \(=\) | \( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8330.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.5153848837\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{7} - x^{6} - 14x^{5} + 6x^{4} + 55x^{3} + 7x^{2} - 36x - 14 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1190) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - x^{6} - 14x^{5} + 6x^{4} + 55x^{3} + 7x^{2} - 36x - 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 16\nu^{3} + 41\nu^{2} - 20\nu - 22 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 18\nu^{3} + 39\nu^{2} - 32\nu - 18 ) / 2 \)
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| \(\beta_{5}\) | \(=\) |
\( 2\nu^{6} - 3\nu^{5} - 26\nu^{4} + 25\nu^{3} + 93\nu^{2} - 35\nu - 48 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{6} - 8\nu^{5} - 66\nu^{4} + 70\nu^{3} + 241\nu^{2} - 110\nu - 128 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 9\beta_{2} + 11\beta _1 + 27 \)
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| \(\nu^{5}\) | \(=\) |
\( -2\beta_{6} + 3\beta_{5} + 11\beta_{4} - 13\beta_{3} + 14\beta_{2} + 55\beta _1 + 28 \)
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| \(\nu^{6}\) | \(=\) |
\( -16\beta_{6} + 18\beta_{5} + 30\beta_{4} - 20\beta_{3} + 79\beta_{2} + 109\beta _1 + 206 \)
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Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.54274 | 1.00000 | 1.00000 | −2.54274 | 0 | 1.00000 | 3.46552 | 1.00000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.02610 | 1.00000 | 1.00000 | −2.02610 | 0 | 1.00000 | 1.10507 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.613936 | 1.00000 | 1.00000 | −0.613936 | 0 | 1.00000 | −2.62308 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.556702 | 1.00000 | 1.00000 | −0.556702 | 0 | 1.00000 | −2.69008 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0.957805 | 1.00000 | 1.00000 | 0.957805 | 0 | 1.00000 | −2.08261 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.65487 | 1.00000 | 1.00000 | 2.65487 | 0 | 1.00000 | 4.04836 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.12679 | 1.00000 | 1.00000 | 3.12679 | 0 | 1.00000 | 6.77683 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8330.2.a.cs | 7 | |
| 7.b | odd | 2 | 1 | 8330.2.a.cr | 7 | ||
| 7.c | even | 3 | 2 | 1190.2.i.o | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1190.2.i.o | ✓ | 14 | 7.c | even | 3 | 2 | |
| 8330.2.a.cr | 7 | 7.b | odd | 2 | 1 | ||
| 8330.2.a.cs | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8330))\):
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\( T_{3}^{7} - T_{3}^{6} - 14T_{3}^{5} + 6T_{3}^{4} + 55T_{3}^{3} + 7T_{3}^{2} - 36T_{3} - 14 \)
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\( T_{11}^{7} - 41T_{11}^{5} - 28T_{11}^{4} + 467T_{11}^{3} + 598T_{11}^{2} - 911T_{11} - 982 \)
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\( T_{13}^{7} - 6T_{13}^{6} - 23T_{13}^{5} + 116T_{13}^{4} - 21T_{13}^{3} - 202T_{13}^{2} + 131T_{13} - 4 \)
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\( T_{19}^{7} - 10T_{19}^{6} - 55T_{19}^{5} + 806T_{19}^{4} - 1536T_{19}^{3} - 4240T_{19}^{2} + 9280T_{19} - 3392 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
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| $3$ |
\( T^{7} - T^{6} + \cdots - 14 \)
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| $5$ |
\( (T - 1)^{7} \)
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| $7$ |
\( T^{7} \)
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| $11$ |
\( T^{7} - 41 T^{5} + \cdots - 982 \)
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| $13$ |
\( T^{7} - 6 T^{6} + \cdots - 4 \)
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| $17$ |
\( (T - 1)^{7} \)
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| $19$ |
\( T^{7} - 10 T^{6} + \cdots - 3392 \)
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| $23$ |
\( T^{7} + T^{6} + \cdots + 92 \)
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| $29$ |
\( T^{7} + 9 T^{6} + \cdots - 153464 \)
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| $31$ |
\( T^{7} - 18 T^{6} + \cdots + 784 \)
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| $37$ |
\( T^{7} - 8 T^{6} + \cdots + 1544 \)
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| $41$ |
\( T^{7} - 15 T^{6} + \cdots - 2192 \)
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| $43$ |
\( T^{7} + 9 T^{6} + \cdots - 223024 \)
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| $47$ |
\( T^{7} - 10 T^{6} + \cdots + 74432 \)
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| $53$ |
\( T^{7} - 10 T^{6} + \cdots - 190768 \)
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| $59$ |
\( T^{7} + 6 T^{6} + \cdots - 25216 \)
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| $61$ |
\( T^{7} - 23 T^{6} + \cdots - 299488 \)
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| $67$ |
\( T^{7} - 13 T^{6} + \cdots - 317952 \)
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| $71$ |
\( T^{7} + 20 T^{6} + \cdots - 75312 \)
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| $73$ |
\( T^{7} - 22 T^{6} + \cdots - 156064 \)
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| $79$ |
\( T^{7} + 8 T^{6} + \cdots + 617724 \)
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| $83$ |
\( T^{7} - 3 T^{6} + \cdots + 502456 \)
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| $89$ |
\( T^{7} - 21 T^{6} + \cdots + 3905432 \)
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| $97$ |
\( T^{7} - 28 T^{6} + \cdots - 216576 \)
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